LECTURE -1 : CLASS VIII : SCIENCE : CHAPTER 4 : MATERIALS : METALS & NON-METALS

CLASS VIII   |    SCIENCE    |    CHAPTER 4

      notes prepared by subhankar Karmakar

                                                                         

• Element:

A substance which cannot be broken down into two or more simpler substances by chemical reactions is is called an element. 

Some of the common elements are:

Hydrogen, helium, carbon, nitrogen, oxygen, sulphur, phosphorus, silicon, chlorine, bromine, iodine, sodium, potassium, magnesium, calcium, zinc, iron, copper, silver, gold and mercury. 

Every element is represented by a symbol. No two elements can have the same symbol.

Symbols of common elements:

1. Hydrogen - H

2. Helium - He

3. Carbon - C

4. Nitrogen - N

5. Oxygen - O

6. Sulphur - S

7. Phosphorus- P

8. Silicon - Si

9. Chlorine - Cl

10. Bromine - Br

11. Iodine - I

12. Sodium - Na

13. Potassium - K

14. Magnesium - Mg

15. Calcium - Ca

16. Zinc - Zn

17. Iron - Fe

18. Copper - Cu

19. Silver - Ag

20. Gold - Au

21. Mercury - Hg

• Atom:

The smallest particle of an element is called atom. An element is a substance which is made up of only one kind of atoms. 

There are as many type of atoms as are elements. So different elements are made up of different kinds of atoms. For example, sulphur element is made up of only sulphur atoms. This means an amount of oxygen is totally made of atoms of oxygen only. 

There are only 92 naturally occurring elements known to us at present. Other elements in the periodic table are synthesized elements. 

Properties of elements:

Different elements have different properties. Some of the most important properties of elements are malleability, ductility, brittleness, lustre, Sonorousness, conductivity, strength, hardness, toughness etc. 

On the basis of their properties, all the elements can be divided into two main groups: 

Metals and nonmetals.

Characteristics of metals:

Metals are malleable and ductile elements. They are good conductors of heat and electricity. Metal are lustrous or shiny. Metals are usually hard and strong. All the metals are solids except Mercury which is a liquid metal. Metals have high densities which means they are heavy. Metals have high melting points and boiling points. Metals are sonorous which means that metals make a ringing sound when we strike them with a hard object. 

Some of the examples of metals are: iron, copper, aluminium, zinc, silver, gold, Platinum, chromium, sodium, potassium, calcium, magnesium, nickel, Cobalt, tin, Mercury, tungsten, manganese, uranium etc. Out of 92 naturally occurring elements, 70 elements are metal.

Characteristics of nonmetals:

Non metals are the elements which are neither malleable nor ductile, they are brittle. Non metals do not conduct heat and electricity. Non metals are not lustrous or shiny. Non metals can be solid, liquid or gases at the room temperature. Non metals have usually low melting points and boiling points. Non metals have low densities which means they are light. Non metals are not sonorous, which means non metals do not make ringing sound when we strike them with a hard object. 

Some of the examples of non metals are: Carbon, sulphur, phosphorus, hydrogen, oxygen, nitrogen, chlorine, fluorine, bromine, iodine, helium, neon, argon, Krypton and xenon. Out of of 92 naturally occurring elements, 22 elements are non metals. Out of these, 10 non metals are solids, 1 non metal is a liquid (bromine), and 11 non metals are gases. 

Metalloids:

There are some elements which show some properties of metals and the other properties of nonmetals. The elements whose properties are intermediate between those of metals and nonmetals are known as metalloids. 

The example of metalloids are: silicon, germanium, arsenic and tellurium.

Physical properties of metals and nonmetals:

a. Malleability:

The property which allows the metals to be hammered into thin sheets is called malleability. Most of the metals are malleable. Gold and silver are the best malleable metals and can be hammered into very fine sheets or foils. Aluminium and copper are also highly malleable. It is due to the property of malleability that metals can be bent to form objects of different shapes by beating with a hammer. 

Non metals are not malleable.

b. Brittleness:

The property due to which non-metals  break on hammering is called brittleness. This means non metals can not be hammered into a thin sheets, it break into small pieces when hammered. All the non metals are brittle.

c. Ductility:

The property which allows the metals to be drawn into wires is called ductility. Ductility is another characteristics property of metals. Generally all the metals are malleable and ductile. 

Gold and silver are among the best ductile metals. Copper and aluminium metals are also very ductile and can be drawn into to tin Copper and aluminium wires.

Non metals are not ductile. 

d. Conductivity:

Heat and electricity can easily flow through metals. Therefore metals are good conductors of heat and electricity. ( as they allow heat and electricity to pass through them easily).

Silver metal is a best conductor of heat. Copper, gold, aluminium and iron metals are good conductors of heat.

Except graphite and diamond all the non metals are bad conductors of heat and electricity. Therefore, they are called insulators. Diamond is a good conductor of heat whereas graphite is a good conductor of electricity. 

e. Lustre:

All the metals have a shiny appearance. This property of metal is known as Lustre of metal. 

All the non metals are not lustrous. Only iodine has a lustre. 

f. Strength:

Metals are usually strong, they have high tensile strength. Metals can hold large weights without snapping. Iron is one of the most strongest material, hence iron is used in construction purposes. 

Non metals are not strong. They have low tensile strength.

g. Sonorousness:

All the metals make a ringing sound when we strike them. This property of metal is known as Sonorousness. Sonorous means capable of producing a ringing sound. 

Non metals are not sonorous. They do not produce ringing sounds when we strike them. 

h. Hardness:

Metals cannot be cut very easily. This property is known as hardness of metal. Only sodium and potassium metals are soft and can be easily cut with a knife. 

Most of the solid non metals are quite soft.

Differences in physical properties of metals and nonmetals

1. Metals: metals are malleable and ductile. 

    Non metals: non metals are neither malleable nor ductile. They are brittle.

2. Metals: metals are good conductors of heat and electricity.

Non metals: nonmetals are poor conductors of heat and electricity except graphite is a good conductor of electricity and diamond is a good conductor of heat.

3. Metals: metals are lustrous. 

Non metals: non metals are not lustrous. They are dull. Only iodine has shiny appearance.

4. Metals: Metals are strong and have a high tension strength. Only sodium and potassium are not strong and have low tensile strength.

Non metals: non metals are not strong. They have a low tensile strength.

5. Metals: metals of sonorous. They make a ringing sound when struck.

Non metals: non metals are not Sonorous. They do not make a ringing sound when struck.

6. Metals: Metals are generally hard. Only sodium and potassium are soft metals.

Non metals: solid nonmetals are quite soft. Only diamond are very hard, in fact it is the hardest material. 

Lecture-1, 2, 3 and 4 : CLASS-X: SCIENCE : Chapter: Reflection of light & concave mirror

CLASS X   |    SCIENCE    |    LIGHT

      Notes prepared by Subhankar Karmakar

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SCIENCE: CLASS X: CBSE (CLASS NOTES)

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CLASS- X; PHYSICS; CHAPTER-1

Topic: REFLECTION OF LIGHT

LECTURE-1

Characteristics of Light:

1. Light is a form of energy.

2. Light always travelled in a straight line which is called Rectilinear Propagation of Light.

3. Light can travel through a medium as well as through vacuum also.

4. Light carries Energy from one point to another point.

5. Physics of light is called Optics.

6. Light exhibits two phenomena called Reflection and Refraction of Light.

REFLECTION OF LIGHT

• When a ray of light travelling in one medium falls on the surface of the second medium and turned back into the first medium, then it is called reflection of light. 

Important characteristics of reflection of light:

• The objects having polished, shining surfaces reflect more light than objects having unpolished, dull surfaces.

• Silver metal is one of the best reflectors of light.

• Ordinary mirrors are made by depositing a thin layer of silver metal on the back side of a plane glass sheet. The Silver layer is then protected by a coat of red paint. The reflection of light in a plane mirror takes place at the Silver surface in it. 

• A ray of light it is the straight line along which light travels.

• A bundle of light rays is called a beam of light.

INCIDENT RAY: The ray of light travelling in one medium when falls on the surface of the second medium is called incident ray.

REFLECTED RAY: The ray of light which after striking the surface of the second medium and turned back into the first medium is called reflected Ray.

POINT OF INCIDENCE: The point where incident ray strikes the surface of the second medium is called point of Incidence.

NORMAL: Perpendicular to the surface of the second medium drawn at the point of Incidence is called Normal.

ANGLE OF INCIDENCE: The angle between incident ray and normal is called angle of incidence. It is denoted by ∠i.

ANGLE OF REFLECTION: The angle between reflected ray and normal is called angle of reflection. It is denoted by ∠r.

LAWS OF REFLECTION:

Reflection of light takes place according to the following two laws:

First law of reflection:

1. The incident ray, the reflected ray and the normal at the point of incidence all lie in the same plane.

Second law of reflection:

2. The angle of incidence is always equal to the angle of reflection, i.e., ∠i = ∠r.

• A ray of light which is incident normally or perpendicularly on a mirror, is reflected back along the same path ( because the angle of incidence as equal as the angle of reflection for such a Ray of light are zero).

• The laws of reflection of light applied to all kinds of mirrors, plane mirrors as well as spherical mirrors.

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LECTURE-2

Regular reflection:

• In regular reflection, a parallel beam of incident light is reflected as a parallel beam in one direction. Regular reflection of light occurs from smooth surfaces like that of a plane mirror or highly polished metal surfaces. A plane mirror always produces regular reflection of light.

Diffuse reflection of light:

• In diffuse reflection, a parallel beam of incident light is reflected in different directions. The diffuse reflection of light takes place from rough surfaces like paper, cardboard, chalk, table, chair and unpolished metal surfaces.

 

Objects: Anything which gives out light rays either its  own or reflected by it, is called an object. For example, a bulb, a candle, a pinhead, an arrow etc. 

Images: Image is an optical appearance produced when light rays coming from an object are reflected from a mirror for refracted through a lens. For example, when we look into a mirror, we see the image of our face, while watching a movie in the cinema hall, we see the images of actors and actresses on the cinema screen. 

Types of images:

There are two types of images

a. Real image 

b. Virtual image

Real image: The image which can be obtained on a screen is called a real image. The image formed on a cinema screen is an example of real image. 

Virtual image: The image which cannot be obtained on a screen is called a virtual image. A virtual image can be seen only by looking into a mirror or a lens. The image of our face in a plane mirror is an example of virtual image. A virtual image is just an illusion. 

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LECTURE-3

Formation of image in a plane mirror: 

Consider a small object O placed in front of a plane mirror MM'. The mirror will form an image I of the object O. The object O gives out light rays OA and OB. OA coming from the object O is incident on the plane mirror at point A and it gets reflected in the direction AX according to the laws of reflection of light, making the angle of reflection r₁ equal to the angle of incidence i₁. Another ray of light OB coming from the object O strikes the mirror at point B and gets reflected in the direction BY, again making the angle of reflection r₂ equal to the angle of incidence i₂. 

The two reflected rays AX and BY are diverging from each other so they cannot meet under left side of the mirror. Let us produce the reflected rays AX and BY backwards. They meet at point I behind the mirror. Now, when the reflected rays AX and BY enter the eye of a person at position E, the eye sees the rays of light in the straight line direction in which the reflected rays enter it. So, the person looking into the mirror from position E sees the reflected rays as if they are coming from the point I behind the mirror. Thus, point I is the image of the object O formed by the plane mirror.

• The nature of image formed by a plane mirror is virtual and erect. The size of the image formed by a plane mirror is equal to that of the object. 

• The image formed in a plane mirror is at the same distance behind the mirror as the object is in front of the mirror.

Lateral inversion: 

• When an object is placed in front of a plane mirror, then the right side of object appears to become the left side of image, and the left side of object appears to become the right side of image. This change of sides of an object and its mirror image is called lateral inversion. 

• The phenomenon lateral inversion is due to the reflection of light. 

Characteristics of an image formed by a plane mirror:

a. The image formed in a plane mirror is virtual. It cannot be received on a screen.

b. The image formed in a plane mirror is erect. It is the same side up as the object.

c. The image in a plane mirror is of the same size as the object.

d. The image formed by a plane mirror is at the same distance behind the mirror as the object is in front of the mirror. 

e. the image formed in a plane mirror is laterally inverted.

Uses of plane mirrors:

a. Plane mirrors are used to see ourselves. The mirrors on our dressing table and in bathroom are plane mirrors.

b. Plane mirrors are fixed on the inside walls of certain shop to make them look bigger.

c. Plane mirrors are fitted at blind turns of some busy roads so that drivers can see the vehicle coming from the other side and prevent accidents.

d. Plane mirrors are used in making periscopes. 

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LECTURE-4

SPHERICAL MIRRORS:

A spherical mirror is a reflecting surface which forms part of a hollow sphere. 

(a) A hollow sphere cut by a plane                          (b) concave mirror  (c) convex mirror

Spherical mirrors are of two types: 

(i) Concave Mirror: A spherical mirror in which the outer bulged surface is silvered polished and the reflection of light takes place from the inner hollow surface is called a concave mirror.

(ii) Convex Mirror: A spherical mirror in which the inner hollow surface is silvered polished and the reflection of light takes place from the outer bulged surface is called convex mirror.

Terms related with Spherical Mirrors

1. POLE: It is the middle point P of the spherical mirror.

2. CENTRE OF CURVATURE: It is the centre C of the sphere of which the mirror forms a part.

3. RADIUS OF CURVATURE: It is the radius ( R= AC or BC) of the sphere of which the mirror forms a part.

4. PRINCIPAL AXIS: The line PC passing through the pole and the centre of curvature of the mirror is called its principal axis.

5. APERTURE: It is the diameter AB of the circular boundary of the spherical mirror.

6. PRINCIPAL FOCUS: A narrow beam of light parallel to the principal axis either actually converges to or appears to diverge from a point F on the principal axis after reflection from the spherical mirror. This point is called the principal focus of the mirror.

CONCAVE MIRROR:

7. Principal focus of a concave mirror:

The principal focus of a concave mirror is a point on its principal axis to which all other light rays which are parallel and close to the axis,  converge after reflection from the concave mirror

8. Focus of a concave mirror:

A concave mirror has a real focus. The focus of a concave mirror is in front of the mirror. 

9. Focal length of a concave mirror: 

The focal length of a concave mirror is the distance between its pole and principal focus. It is denoted by f. 

10. Relation between radius of curvature (R) and focal length (f) of a spherical mirror:

The focal length of a spherical mirror is equal to half of its radius of curvature. It is true for both concave and convex mirror. 

∴ f = R/2

Principal focus and focal length of a convex mirror: 

• The principal focus of a convex mirror is a point on its principal axis from which a beam of light rays, initially parallel to the axis, appears to diverge after being reflected from the convex mirror. 

• A convex mirror has virtual focus, focus of a convex mirror is situated behind the mirror. 

Lecture-3 : CLASS-X: SCIENCE : Chapter: Electricity

CLASS X: SCIENCE: ELECTRICITY

notes prepared by Subhankar Karmakar

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ELECTRIC CHARGE: (Q)

Electric charge is the property that causes electrically charged particles to attract or repel each other.

There are two types of electric charge, positive and negative. Protons are positively charged particles, while electrons are negatively charged particles. Neutrons have no electric charge.

Opposite charges attract each other, while like charges repel each other. The strength of the electric force between two charged particles depends on the magnitude of the charges and the distance between them. The greater the charge and the closer the distance, the stronger the electric force between them.

Electric charge is measured in coulombs (C). The unit of electric charge is the amount of charge that flows through a wire when a current of one ampere flows for one second.

Electric charge plays a fundamental role in many areas of science and technology, including electricity, electronics, magnetism, and chemistry.

ELECTRIC CURRENT: (I)

Electric charge flows through a wire if potential difference is applied between the end of the wire. 

When electric charge flows through a wire , it is called electric current. Therefore, the electric current is a flow of electric charges in a conductor such as a metal wire. 

The magnitude of electric current in a conductor is the amount of electric charge passing through a given point of the conductor in one second. 

If a charge of Q coulombs flows through a conductor in time t seconds, then the magnitude I of the electric current flowing through it is given by:

FORMULA OF ELECTRIC CURRENT:

Current, I = Q/t

UNIT OF ELECTRIC CURRENT:

The SI unit of electric current is ampere. It is denoted by the symbol A. 

DEFINITION OF 1 AMPERE CURRENT:

1 A current : When 1 coulomb of charge flows through any cross section of a conductor  in 1 second, the electric current flowing through it is said to be 1 ampere. 

1 ampere = 1 coulomb / 1 second

Or, 1 A = 1 C/1 s

There are two smaller unit of current called

1. milliampere (mA) and 2. microampere (μA)

1 milliampere (1 mA)= 1/1000 A = 10⁻³ A 

1 microampere (1 μA) = 1/1000000 A = 10⁻⁶ A

AMMETER:

Electric current is measured by an instrument called ammeter. An ammeter is always connected in series with the circuit in which the current is to be measured. Since the entire current passes through the ammeter, therefore, an ammeter should have very low resistance, so that it may not change the value of the current flowing in the circuit. 

CONTINUOUS FLOW OF ELECTRIC CURRENT:

If we maintain a steady potential difference between the two ends of a conductor so as to get a continuous flow of current is to connect the conductor between the terminals of a cell or a battery.

DIRECTION OF ELECTRIC CURRENT:

The conventional direction of electric current is from positive terminal of a cell or a battery to the negative terminal through the outer circuit. The actual flow of electrons which is really responsible for the electric current is however from negative terminal to positive terminal of a cell which is opposite to the direction of conventional current.

Electric current flows through a conductor due to the presence of free electron in the conductor. The electron which can move easily through a conductor is known as a free electron. 

ELECTRIC CIRCUITS:

A continuous conducting path consisting of wires and other resistances like electric bulb and a switch, between the two terminals of a cell or a battery along which an electric current flows is called an electric circuit.

Symbols for Electrical Components:

a. Cell : It supplies a continuous supply of potential difference.

b. Battery or Combination of cells:

c. Connecting Wire: It is made of conductors with a covering of insulator through which electric current can flow

d. A Wire Joint: A joint where two connecting wire is joined together. 

e. Wires crossing without connection:

When two wires cross each other without touching each other

f. Fixed Resistance or Resistor:

When the value of the resistor does not change.

g. Variable Resistance or Rheostat:

When the value of the resistor can change with time.

h. Ammeter: instrument which measures current.

i. Voltmeter: instrument which measures potential difference.

j. Galvanometer: instrument which detects the flow of current through a wire.

k. An open switch: which prevents the flow of current through a circuit.

l. A closed switch: which enables electric current to flow through a circuit.

m. Electric bulb: which emits light when electric current flow through it.

Circuit Diagrams:

A diagram which indicates how different components in a circuit have been connected by using the electrical symbols for the components, is called a circuit diagram. 

A simple electric circuit. In this circuit, a resistor R has been connected to the two terminals of a cell through a switch (which is closed). An ammeter A has been put in series with the resistor R. This is to measure current in the circuit. A voltmeter V has been connected across the ends of the resistor R, that is, voltmeter is connected in parallel with the resistor. This voltmeter is used to measure potential difference or voltage across the ends of the resistor R.

LECTURE -4 : CLASS VIII : SCIENCE : CHAPTER 3 : SYNTHETIC FIBRES & PLASTICS

CLASS VIII   |    SCIENCE    |    CHAPTER 3

      notes prepared by subhankar Karmakar

                                                                         

PLASTICS:

A Plastic is a synthetic material which can be moulded into desired shape when soft and then hardened to produce a durable article. Plastics are also polymers. Plastics are made from petroleum products called petrochemicals. 

Some of the examples of plastics are: Polythene, polyvinyl chloride (PVC), bakelite, melamine and Teflon. Nylon is also a plastic.

Polythene: 

It is a plastic obtained by the polymerization of a chemical compound known as ethene. Polythene is tough and durable. Polythene is used in making Polythene bags, waterproof plastic sheets, bottles, buckets and dustbins. Polythene is also used for packaging.

Polyvinyl chloride (PVC): 

It is a strong and hard plastic. It is not as flexible as polythene. PVC is used for making insulation of electric wires, pipes, garden hoses, rain coats, seat covers etc.

Bakelite: 

It is a very hard and tough plastic. Bakelite is a poor conductor of heat and electricity. Bakelite is used for making the handles of various cooking utensils. 

Bakelite is used for making handles cooking utensils because

1. It is a poor conductor of heat and

2. It does not become soft on getting heated as it is a thermosetting plastic.

Bakelite is also used for making electrical fittings such as electric switches, plugs and sockets etc. because

1. It does not conduct electricity and

2. It does not become soft on getting heated.

Melamine: 

It is a plastic which can tolerate heat better than other plastics and resists fire. 

Melamine is used for making floor tiles, unbreakable kitchenwares, ashtrays and fire resistant fabrics. 

Melamine is a fire resistant plastic, hence, the uniforms of fire man heavy coating of melamine plastic to make them fire resistant. 

Special plastic cookwares made of melamine is used in microwave oven for cooking food. In microwave oven the heat cooks the food but does not affect the plastic vessel.

Teflon:

Teflon is a special plastic on which oil and water do not stick. Oil and water do not stick on Teflon plastic because it has a slippery surface. Teflon also withstands high temperature. Teflon is used for giving non stick coating on cookwares like non stick frying pans. Teflon is also used for making soles of electric irons.

TYPES OF PLASTICS:

There are two types of plastics

1. Thermoplastics

2. Thermosetting plastics

a. Thermoplastics:

A Plastic which can be softened repeatedly by heating and can be moulded in two different shapes again and again, is called a thermoplastic. 

Properties of thermoplastics:

Thermoplastics are flexible so they can be bent easily without breaking. Thermoplastics are also known as thermosoftening plastics. Some of the examples of thermoplastics are: polythene and polyvinyl chloride (PVC).

Uses of thermoplastics:

Thermoplastics are used for making those articles which do not get too hot and are flexible. Thermoplastics are used for making insulation of electric wires and cables, various types of plastic containers like plastic bottles, plastic jars, combs, toys, plastic bags, raincoats, seat covers, bristles of brushes, packaging materials and chairs. 

Thermoplastics are used for making the insulation of electric wires because

1. They do not conduct electricity, and

2. They are flexible. 

b. Thermosetting plastics:

A plastic which once set, does not become soft on heating and cannot be moulded a second time, is called a thermosetting plastic. Once Seth in a given shape and solidified, a thermosetting plastic cannot be re-softened or remoulded. Thus, and article or object made of thermosetting plastic will retain its original shape permanently, even on heating. Thermosetting plastics are also known as thermosets.

Some of the examples of thermosetting plastics are: bakelite and melamine. 

Properties of thermosetting plastics:

Most of the thermosetting plastics hard and rigid. Thermosetting plastics are not flexible. Therefore, thermosetting plastics cannot bend. When an article made of thermosetting plastic is forced to bend, it breaks. Thermosetting plastics do not become soft on heating. Thermosetting plastics are used for making those articles which may get too hot during use and are hard and rigid so that they do not bend at all. 

Uses of thermosetting plastics:

Thermosetting plastics are used for making handles of cooking utensils, plates, cups, floor tiles, electrical fittings like electrical switches, plugs and sockets, ballpoint pens and telephone instruments. 

Reasons of using thermosetting plastics for making the handles of cooking utensils:

Thermosetting plastics are used for making the handles of cooking utensils because

a. They do not softened on getting heated, and

b. They are poor conductors of heat.

Reasons of using thermosetting plastics for making electrical fittings:

Thermosetting plastics are used for making electrical fittings such as electric switches, plugs and sockets because

a. They do not become soft on getting heated, and

b. They do not conduct electricity.

Plastics use for making toothbrush:

The handle and bristles of a toothbrush cannot be made of the same plastics, because the handle of a toothbrush has to be hard and rigid whereas the bristles of a toothbrush have to be soft and flexible. This means that the handle of a toothbrush should be made of thermosetting plastic whereas its bristles should be made of thermoplastic.

Structures of thermoplastics and thermosetting plastics:

In thermoplastics long polymer chains are not cross linked with one another but in thermosetting plastics the long polymer chains are cross linked with one another.

As in thermoplastic long polymer chains are not cross-linked, hence, the individual polymer chains can slide over one another and thermoplastic material become soft and ultimately melts.

As in thermosetting plastics, long polymer chains are cross linked, these cross-links prevent the displacement of individual polymer chains on being heated. Due to this, thermosetting plastics do not become soft on heating once they have been set into to a particular shape.

Useful properties of plastics:

Some of the most useful properties of plastics are as follows. 

a. Plastics are chemically unreactive. 

Plastics do not react with air and water. Therefore, plastics are resistant to corrosion. Due to this property, plastic containers are used to store various kinds of materials, including many chemicals.

b. Plastics are bad conductors of heat and electricity:

Plastics neither conducts heat nor electricity. They are used as insulators. This properties make it the material for making of handles of the cooking utensils and electrictical fittings like switches, plugs and sockets as well as it is used as the covering of the electrical wires. 

c. Plastics can be moulded into different shapes.

Since plastics can be very easily moulded, they are used to make a large variety of articles like buckets, mugs, furniture, bags, sheets, slippers, electrical fittings, toys, combs, toothbrushes etc.

d. Plastics are quite cheap and easily made:

Plastics a generally cheaper than metals and wood. Plastics production is also easy comparing to the extraction and purification of metals. These property made plastics as the first choice for making many of the household as well as industrial articles. It replaces metals in most of the cases. 

e. Plastics are light, strong and durable:

As the plastics has low density, they are lighter than metals. Plastics has good strength and they are durable too. Due to their low cost, high strength, easy availability, lightweight, long durability and corrosion resistant properties, plastics are now-a-days widely used for the making of most of the articles we use. 

BIODEGRADABLE & NON-BIODEGRADABLE MATERIALS:

A material which gets decomposed through natural processes such as action of bacteria and other other natural factors are called biodegradable material. They are environment friendly. Plant and animal wastes, paper, cotton, cloth, wool, jute, wood are some examples of biodegradable materials. 

A material which is not easily decomposed by natural processes are called non-biodegradable material. They are not environment friendly. They directly or indirectly pollute nature. Plastics, glass, metal foils, aluminium cans are some examples of non-biodegradable materials. 

PLASTICS POLLUTE ENVIRONMENT:

Plastic materials have a bad effect on the environment. The excessive use of plastic materials affect our environment in the follwing ways. 

a. Plastics are non-biodegradable, so, the waste plastic articles keep on accumulating in the surroundings and pollute the environment badly.

b. The waste plastic articles thrown in the drain choked them and causes them to overflow and create an unhygienic condition. 

c. Plastics may be eaten by the animals, which causes permanent damage to the animals. 

d. If plastic materials are burnt they produce toxic gases which pollutes air and ultimately become the causes of many diseases.

The ways to prevent environment pollution from excessive plastic wastes:

Plastics are very useful materials but use of plastic articles is not good for the environment. Therefore, some steps should be taken to save the environment from plastic waste. They are as follows:

1. We should try to reduce or minimise the use of plastic by using other alternative materials. Like instead of using Polythene bags we can use jute or paper bags.

2. We should not throw Polythene bags, wrappers of chips, biscuits or other eatables in water bodies, on the roads, in parks or picnic places. We should use dustbin for solid waste.

3. We should reuse the plastic containers which come with jams, pickles, oils and other packed food. 

4. Plastic wastes should be recycled. All the plastic waste in in the different places should be collected and sent for recycling to plastic making factories. In plastic factories, the waste plastic articles are melted and used to make new plastic articles. All the recycled plastic product are given certain colours, so that buyers can understand they are recycled plastic product and they don't use them for stories of food.

5. We should follow 3R principle. It means Reduce, Reuse and Recycle for plastic products.

LECTURE: 3 : CLASS XI: PHYSICS : UNITS & MEASUREMENTS

CLASS XI   |    PHYSICS    |    CHAPTER 2

      notes prepared by subhankar Karmakar

                                                                                  

Accuracy and precision:

Accuracy: it refers to the closeness of a measurement to the true value of the the physical quantity. It indicates the relative freedom from errors. As we reduce the errors, the measurement becomes more accurate.

Precision: it refers to the resolution or the limit to which the quantity is measured. Precision is determined by the least count of the measuring instrument. The smaller the list count, greater is the precision. 

Errors in measurement: 

The error in a measurement is equal to the difference between the true value and the measured value of the quantity.

      Error = true value - measured value

Different types of errors:

1. Constant errors

2. Systematic errors

     a. Instrumental errors

     b. Imperfection in experimental technique

     c. Personal errors

     d. Errors due to external causes

3. Random errors

4. Least count error

5. Gross errors or mistakes

1. Constant errors: the errors which affect each observation by the same amount are called constant errors. 

2. Systematic errors: the errors which tend to occur in one direction, either positive or negative, are called systematic errors. Systematic errors are classified as follows:

     a. Instrumental errors: These errors occur due to the inbuilt defect of the measuring instrument. 

     b. Imperfections in experimental technique: These errors are due to the limitations of the experimental arrangement. 

     c. Personal errors: These errors arise due to to individual's bias, lack of proper setting of apparatus or individual's carelessness in taking observations without observing proper precautions.

    d. Errors due to external causes: These errors arise due to the the change in external conditions.

3. Random errors: The errors which occur irregularly and at random, in magnitude and direction, are called random errors.

4. Least count error: This error is due to the limitation imposed by the the least count of the measuring instrument.

5. Gross errors or mistakes: These errors are due to either carelessness of the person or due to improper adjustment of the apparatus.

Different types of error measurement: 

a. True value of a physical quantity: arithmetic mean of all the measurements can be taken as the true value of the measured quantity. 

If a₁, a₂, a₃, a₄, a₅ ...... aₙ be the n measured values of a physical quantity, then is true value 

aₘₑₐₙ or ā = (a₁+ a₂ + a₃ + a₄ + a₅ +......+ aₙ )/n

b. Absolute Error: The magnitude of the difference between the true value of the quantity measured and the individual measured value is called absolute error. 

|∆a₁| = |ā - a₁|

|∆a₂| = |ā - a₂|

|∆a₃| = |ā - a₃|

............................

|∆aₙ| = |ā - aₙ|

c. Mean or final absolute error:

The arithmetic mean of the positive magnitudes of all the absolute errors is called mean absolute error. It is given by

∆ā = (|∆a₁|+ |∆a₂| + |∆a₃|  +......+ |∆aₙ| )/n

The final result of the measure of a physical quantity can be expressed as

    a = ā ± ∆ā

d. Relative error:

The ratio of the mean absolute error to the true value of the measured quantity is called relative error. 

Relative error, δa = ∆ā /ā

e. Percentage error:

The relative error expressed in percent is called percentage error. 

Percentage Error = (∆ā/ā) x 100%

COMBINATION OF ERRORS:

a. Error in the sum of two quantities:

Let ∆A and ∆B be the absolute errors in the two quantities A and B respectively. Then, 

Measured value of A = A ± ∆A

Measured value of B = B ± ∆B

Consider the sum, Z = A + B

The error ∆Z in Z is then given by 

Z ± ∆Z = (A ± ∆A) + (B ± ∆B)

            = (A + B) ± (∆A + ∆B)

            = Z ± (∆A + ∆B)

∴ ∆Z = (∆A + ∆B)

b. Error in the difference of two quantities

Consider the difference, Z = A - B

The error ∆Z in Z is then given by 

Z ± ∆Z = (A ± ∆A) - (B ± ∆B)

            = (A - B) ± ∆A ∓ ∆B

            = Z ± ∆A ∓ ∆B

For error ∆Z to be maximum, ∆A and ∆B must have the same sign, therefore

∴ ∆Z = (∆A + ∆B)

c. Error in the product of two quantities:

Consider the product , Z = AB

The error ∆Z in Z is given by

Z ± ∆Z = (A ± ∆A)(B ± ∆B)

            = AB ± A∆B ± B∆A ± ∆A. ∆B

Dividing LHS by Z and RHS by AB [∵ Z = AB]

1 ± ∆Z/Z = 1 ± ∆B/B ± ∆A/A ± (∆A/A)(∆B/B)

As the last term is very small, it can be neglected. 

 ± ∆Z/Z =  ± (∆B/B + ∆A/A)

∴ ∆Z/Z =  (∆B/B + ∆A/A)

d. Error in the division or quotient

Consider the product , Z = A/B

The error ∆Z in Z is given by

Z ± ∆Z = (A ± ∆A)/(B ± ∆B)

            = A(1 ± ∆A/A)/{B(1 ± ∆B/B)}

            = (A/B)(1 ± ∆A/A)(1 ± ∆B/B)⁻¹

            = Z(1 ± ∆A/A)(1 ∓ ∆B/B)

 [∵ (1 + x)⁻¹ ≃ 1 + nx when x <<1]

Dividing both sides by Z 

1 ± ∆Z/Z = 1 ∓ ∆B/B ± ∆A/A ± (∆A/A)(∆B/B)

As the last term is very small, it can be neglected. 

∴ ∆Z/Z =  (∆B/B + ∆A/A)

e. 1. Error in the power of a quantity:

Consider. Z = Aⁿ

The error ∆Z in Z is given by

Z ± ∆Z = (A ± ∆A)ⁿ = Aⁿ (1 ± ∆A/A)ⁿ

                                  = Z (1 ± n∆A/A)

[∵ (1 + x)⁻¹ ≃ 1 + nx when x <<1]

Dividing both sides by Z, we get

1 ± ∆Z/Z = 1 ± n(∆A/A)

or  ∆Z/Z = n(∆A/A)

   2. General rule:

   Consider. Z = Pᵃ Qᵇ / Rᶜ

Then ∆Z/Z = a(∆P/P) + b(∆Q/Q) + c(∆R/R)

Numericals :

Q1. The length of a rod as measured in an experiment was found to be 2.48 m, 2.46 m, 2.49 m, 2.50 m, 2.48 m. Find the (a) average length, (b) the absolute error in each observation and (c) the percentage error.

Soln. (a) Average length 

= (2.48 + 2.46 + 2.49 + 2.50 + 2.48)/5

= 12.41/5 = 2.482 = 2.48

∴ true length, ā = 2.48 m

(b) The absolute errors in different measurements are:

|∆a₁| = |ā - a₁| = |2.48 - 2.48| = 0.00 m

|∆a₂| = |ā - a₂| = |2.48 - 2.46| = 0.02 m

|∆a₃| = |ā - a₃| = |2.48 - 2.49| = 0.01 m

|∆a₄| = |ā - a₄| = |2.48 - 2.50| = 0.02 m

|∆a₅| = |ā - a₅| = |2.48 - 2.48| = 0.00 m

(c) the absolute error, |∆ā| 

= (0.00 + 0.02 + 0.01+ 0.02 + 0.00)/5

= 0.01 m

∴ correct length, ā ± |∆ā| = 2.48 ± 0.01 m

∴ percentage error = (0.01/2.48)x 100%

                                 = 0.40%

Q2. In successive measurements, the readings of the period of oscillation of a simple pendulum were found to be 2.63 s, 2.56 s, 2.42 s, 2.71 s and 2.80 s in an experiment. Calculate (a) mean value of the period of oscillation

(b) absolute error in each measurement

(c) mean absolute error

(d) relative error

(e) percentage error and

(f) Express the result in proper form.

Soln. (a) mean period of oscillation 

= (2.63 + 2.56 + 2.42 + 2.71 + 2.80)/5

= 13.12/5 = 2.624 s ≃ 2.62 s

(b) absolute errors in different measurement,

|∆a₁| = |ā - a₁| = |2.62 - 2.63| = 0.01 s

|∆a₂| = |ā - a₂| = |2.62 - 2.56| = 0.06 s

|∆a₃| = |ā - a₃| = |2.62 - 2.42| = 0.20 s

|∆a₄| = |ā - a₄| = |2.62 - 2.71| = 0.09 s

|∆a₅| = |ā - a₅| = |2.62 - 2.80| = 0.18 s

(c) mean absolute error, |∆ā|

= (0.01 + 0.06 + 0.20 + 0.09 + 0.18)/5

= 0.11 s

(d) relative error δā = |∆ā|/ā

= 0.11/2.62 = 0.04

(e) percentage error = 0.04 x 100% = 4%

(f) in terms of absolute error, 

(2.62 ± 0.11) s

In terms of percentage error, 

(2.62 ± 4%) s.

Homework:

Q3. In an experiment, refractive index of glass was observed to be  1.45, 1.56, 1.54, 1.44, 1.54 and 1.53. Calculate (a) mean value of refractive index, (b) mean absolute error, (c) fractional error aur relative error, (d) percentage error, 

(e) express the result in terms of absolute error and percentage error.

Q4. In an experiment to measure focal length of a concave mirror, the value of focal length in successive observations turns out to be 17.3 cm, 17.8 cm, 18.3 cm, 18.2 cm, 17.9 cm and 18.0 cm. Calculate the mean absolute error and percentage error. Also, express the result in a proper way. 

Numericals on combination of errors:

Q5. Two resistances R₁ = 100 ± 3 Ω and R₂ = 200 ± 4 Ω are connected in series. What is their equivalent resistance?

Q6. Two different masses are determined as (23.7 ± 0.5) g and (17.6 ± 0.3) g. What is the sum of their masses?

Q7. The initial and final temperatures of a water bath are (18 ± 0.5)°C and (40 ± 0.3)°C. What is the rise in temperature of the bath?

Q8. The resistance R =V/I, where V = 100 ± 5 V and I = 10 ± 0.2 A. Find the percentage error in R.

Q9. The percentage errors in the measurement of mass and speed are 2% and 3% respectively. How much will be the maximum error in the estimate of kinetic energy obtained by measuring mass and speed?

Q10. The length, breadth and height of a rectangular block of wood were measured to be :

l = 12.13 ± 0.02 cm;

b = 8.16 ± 0.01 cm and

h = 3.46 ± 0.01 cm

Determine the percentage error in the volume of the block.

Q11. The period of oscillation of a simple pendulum is T = 2π √(L/g). Measured value of L is 20.0 cm known to 1 mm accuracy and time for 100 oscillations of the pendulum is found to be 90 s using a wrist watch of 1 s resolution. What is the accuracy in the determination of g?

Q12. A physical quantity X is given by 

X = (a²b³)/(c√d). If the percentage errors of measurement in a, b, c and d are 4%, 2%, 3% and 1% respectively, then calculate the percentage error in X.

    

LECTURE -3 : CLASS VIII : SCIENCE : CHAPTER 3 : SYNTHETIC FIBRES & PLASTICS

CLASS VIII   |    SCIENCE    |    CHAPTER 3

      notes prepared by subhankar Karmakar

                                                                         

ADVANTAGES OF SYNTHETIC FIBRES

Important properties and advantages of synthetic fibres as follows:

1. Synthetic fibres are very strong whereas natural fibres like cotton, wool and silk have low strength. 

2. Synthetic fibres are more durable. Synthetic fibres have high resistance to abrasion and hence, the clothes made of synthetic fibres are very durable whereas natural fibres like cotton, wool and silk have low abrasion resistance due to which the clothes made of natural fibres are not much durable. They do not last long.

3. Synthetic fibres absorb very little water and hence, the clothes made of synthetic fibres dry up quickly. On the other hand, natural fibres like cotton, wool and silk absorb a lot of water. So, the clothes made a natural fibres do not dry up quickly.

4. Synthetic fibres are wrinkle resistant and hence, class made of synthetic fibres do not get crumpled easily during washing or wear. They keep permanent creases. But, natural fibres like cotton, wool and silk are not wrinkle resistant. So, the clothes made of natural fibres get crumpled easily during washing and wear.

5. Synthetic fibres are quiet lightweight. Whereas natural fibres are comparatively heavy.

6. Synthetic fibres are extremely fine. So the fabrics made from synthetic fibres have a very smooth texture. But, natural fibres are not so fine. Therefore, the fabrics made from natural fibres do not have a very smooth texture. 

7. Synthetic fibres are not attacked by moths. But natural fibres are damaged by moths.

8. Synthetic fibres do not shrink. So, the clothes made of synthetic fibre retain their original size even after washing. On the other hand, natural fibres shrink after washing.

9. Synthetic fibres are less expensive and readily available as compared to natural fibres.

10. Clothes made from synthetic fibres are easier to maintain as compared to those made from natural fibres.

11. The manufacturing of fully synthetic fibres is helping in the conservation of forests. As the synthetic fibres are made from petrochemical products, so no trees have to be cut down for making them. On the other hand, semi synthetic fibres like Rayon are made from wood pulp request cutting down of forest trees.

DISADVANTAGES OF SYNTHETIC FIBRES:

There are several disadvantages of synthetic fibres. They are as follows:

1. Synthetic fibres always melt on heating. Therefore, if a person is wearing clothes made of synthetic fibres and his clothes catch fire accidentally, then the synthetic fibres of clothes melt and stick to the body of the person causing severe burns. So, it is quite safe to wear clothes made of natural fibres while working in the kitchen or in a science laboratory.

2. The clothes made of synthetic fibres are not suitable for wearing during hot summer weather. As the synthetic fibres are extremely fine so the clothes made of synthetic fibres do not have sufficient pores for the sweat to come out, evaporate and coo our body. Due to this, clothes made of synthetic fibres make us feel hot and uncomfortable during summer. 

LECTURE: 2 : CLASS XI: PHYSICS : UNITS & MEASUREMENTS

CLASS XI   |    PHYSICS    |    CHAPTER 2

      notes prepared by Subhankar Karmakar

Conversion table from degree to radian:

a.   1° = 1.745 x 10⁻² rad

b.   1' = 2.91 x 10⁻⁴ rad

c.   1" = 4.85 x 10⁻⁶ rad

 

Q1. The moon is observed from two diametrically opposite points A and B on the earth. The angle θ subtended at the moon by the two directions of observation is 1°54'. Given the diameter of the earth to be 1.276 x 10⁷ m, compute the distance of the Moon from the Earth. 

 

Soln. Here the parallactic angle 

θ = 1°54' = 1.745 x 10⁻² + 54 x 2.91 x 10⁻⁴ rad

                = 3.32 x 10⁻² rad.

Here, b = AB = 1.276 x 10⁷ m

The distance of the Moon from the Earth,

S = b/θ = 1.276 x 10⁷/3.32 x 10⁻²

             = 3.84 x 10⁸ m

 

Q2. The angular diameter of the sun is 1920". If the distance of the sun from the earth is 1.5 x 10¹¹ m, what is the linear diameter of the sun?

 

Soln. Distance of the sun from the earth

          S = 1.5 x 10¹¹ m 

          Angular diameter of the sun

          θ = 1920" = 1920 x 4.85 x 10⁻⁶ rad

                           = 9312 x 10⁻⁶ rad

Linear diameter of the sun

          D = Sθ = 1.5 x 10¹¹ x 9312 x 10⁻⁶ m

                           = 13968 x 10⁵ m

                           = 1.4 x 10⁶ km

 

 

• DIMENSION OF A PHYSICAL QUANTITY:

 

All the derived physical quantities can be expressed in terms of some combination of the seven fundamental or base quantities. We call these fundamental quantities as the seven dimensions of the world, which are denoted with square brackets [ ]. 

 

• Dimension of length = [L]

• Dimension of mass = [M]

• Dimension of time = [T]

• Dimension of electric current = [A]

• Dimension of thermodynamic temperature = [K]

• Dimension of luminous intensity = [cd]

• Dimension of amount of substance = [mol]

 

The dimensions of a physical quantity are the powers to which the fundamental quantities must be raised to represent that quantity completely. 

For example, 

Density = Mass/Volume = Mass/ (Length x breadth x height) 

Dimensions of density = [M]/([L] x [L] x [L])

= [M¹L⁻³T⁰]

 

·        Area = [M⁰L²T⁰] = m²

·        Volume = [M⁰L³T⁰] = m³

·        Density = [M¹L⁻³T⁰] = kg m⁻³

·        Speed or Velocity = [M⁰L¹T⁻¹] = m/s

·        Acceleration = [M⁰L¹T⁻²] = m/s²

 

DIFFERENT TYPES OF VARIABLES AND CONSTANTS: 

 

There are two types of variables

1. Dimensional variables: 

 

The physical quantities which possess dimensions and have variable values are cal dimensional variables. For example, area, volume, velocity, force, power, energy etc.

 

2. Dimensionless variables: 

 

The physical quantities which have no dimensions but have variable values are called dimensionless variables. For example, angle, specific gravity, strain etc.

 

There are two types of constants:

 

1. Dimensional constants: 

 

The physical quantities which possess dimensions and have constant values are called dimensional constants. For examples, gravitational constant, Planck's constant, electrostatic constant etc.

 

2. Dimensionless constants: 

 

The constant quantities having no dimensions are called dimensionless constants. For example, π, e etc. 

 

Application of dimensional analysis: 

 

The method of studying a physical phenomenon on the basis of dimensions is called dimensional analysis. 

 

Following are the three main uses of dimensional analysis: 

 

1. To convert a physical quantity from one system of units to other. 

2. To check the correctness of a given physical relation.

3. To derive a relationship between different physical quantities.

 

1. Conversion of one system of units to other:

 

As the magnitude of physical quantities remain same and does not depend upon our choices of units, therefore, 

                   Q = n₁u₁ = n₂u₂

where Q is the magnitude of the physical quantity, u₁ and u₂ are the units of measurement of that quantity and n₁ and n₂ are the corresponding numerical values. 

u₁ = M₁^aL₁^bT₁^c

u₂ = M₂^aL₂^b T₂^c

∴ n₁[M₁^aL₁^bT₁^c] = n₂[M₂^aL₂^b T₂^c]

⟹  n₂ = n₁ [M₁/M₂]^a [L₁/L₂]^b [T₁/T₂]^c   

 

Q1. Convert 1 Newton into dyne.

 

Soln. Newton is the SI unit of force and dyne the CGS unit of force. Dimensional formula of force is M¹L¹T⁻²

∴ a = 1, b = 1, c = -2

In SI system; 

M₁ = 1 kg = 1000 g

L₁ = 1 m = 100 cm

T₁ = 1 s and n₁ = 1 (Newton)

In CGS system;

M₂ = 1 g ; L₂ = 1 cm ; T₂ = 1 s

∴ n₂ = n₁ [M₁/M₂]^a [L₁/L₂]^b [T₁/T₂]^c 

         = 1 x [1000/1]¹ x [100/1]¹ x [1/1]⁻²

        = 1 x 10³ x 10²

        = 10⁵ 

∴ 1 N = 10⁵ dyne

 

Q2. Convert 1 erg into Joule.

 

Soln. Erg is CGS unit of energy whereas joule is SI unit of energy. Dimensional formula of energy is M¹L²T⁻².

∴ a = 1, b = 2, c = -2

In CGS system;

M₁ = 1 g ; L₁ = 1 cm ; T₁ = 1 s ; n₁ = 1

In SI system; 

M₂ = 1 kg = 1000 g

L₂ = 1 m = 100 cm

T₂ = 1 s and n₂ = ?

∴ n₂ = n₁ [M₁/M₂]^a [L₁/L₂]^b [T₁/T₂]^c 

         = 1 x [1/1000]¹ x [1/100]² x [1/1]⁻²

        = 1 x 10⁻³ x 10⁻⁴

        = 10⁻⁷

∴ 1 erg =  10⁻⁷ N

 

Q3. The density of Mercury is 13.6 g/cm³ in CGS system. Find its value in SI system.

 

Soln. The dimensional formula of density is

M¹L⁻³T⁰

∴ a = 1, b = - 3, c = 0

In CGS system;

M₁ = 1 g ; L₁ = 1 cm ; T₁ = 1 s ; n₁ = 13.6

In SI system; 

M₂ = 1 kg = 1000 g

L₂ = 1 m = 100 cm

T₂ = 1 s and n₂ = ?

∴ n₂ = n₁ [M₁/M₂]^a [L₁/L₂]^b [T₁/T₂]^c 

        = 13.6 x [1/1000]¹ x [1/100]⁻³ x [1/1]⁰

        = 13.6 x 10⁻³ ⁺ ⁽⁻²⁾⁽⁻³⁾ 

        = 13.6 x 10³

∴ The density of Mercury in SI unit is 13.6 x 10³ kg/m³

 

Q4. If the value of atmospheric pressure is 10⁶ dyne / cm², find its value in SI units.

 

Q5. If the value of universal gravitational constant in SI unit is 6.6 x 10⁻¹¹ N m² kg⁻², then find its value in CGS unit.

 

 

 

2. CHECKING THE DIMENSIONAL CONSISTENCY OF EQUATIONS:

 

• Principle of homogeneity of dimensions:

 

According to this principle, a physical equation will be dimensionally correct if the dimensions of all the terms occurring on both side of the equation are the same. 

 

Q6. Check the dimensional accuracy of the equation of motion s = ut + ½at².

 

Soln. Dimensions of different terms are

[s] = [L],

[ut] = [LT⁻¹] x [T] = [L],

[½at²] =  [LT⁻²] x [T²] = [L]

 

As all the terms on both sides of the equation have the same dimensions, show the given equation is dimensionally correct. 

 

Q7. Check the correctness of the equation

       FS = ½mv² - ½mu²

       Where F is a force acting on a body of mass m and S is the distance moved by the body when its velocity changes from u to v.

 

Soln. 

    [FS] = [M¹L¹T⁻²][L] = [M¹L²T⁻²]

    [½mv²] = [M][LT⁻¹]² = [M¹L²T⁻²]

    [½mu²] = [M][LT⁻¹]² = [M¹L²T⁻²]

Since the dimensions if all the terms in the given equation are same, hence the given equation is dimensionally correct. 

 

Q8. The Vander Waal's equation for a gas is

       ( P + a/V²)(V - b) = RT

Determine the dimensions of a and b. Hence write the SI units of a and b.

 

Soln. Since the dimensionally similar quantities can be added or subtracted, therefore, 

[P] = [a/V²] 

⟹ [a] = [ PV²] = [ M¹L⁻¹T⁻²] [L³]² = [M¹L⁵T⁻²]

Also, [b] = [V] = [L³]

∴ The SI unit of a is kg m⁵/s² and that of b is m³

 

3. DEDUCING RELATION AMONG THE PHYSICAL QUANTITIES:

 

By making use of the homogeneity off dimensions, we can derive an expression for a physical quantity if we know the various factors on which it depends

 

Q9. Derive an expression for the centripetal force F acting on a particle of mass m moving with velocity v in a circle of radius r.

 

Soln. Centripetal force F depends upon mass M, velocity V and radius r.

Let F ∝ mᵃ vᵇ rᶜ 

∴ F = K mᵃ vᵇ rᶜ --------(1)

where K is a dimensionless constant. 

Dimensions of the various quantities are

 [m] = [M],  [v] = [LT⁻¹],  [r] = [L]

Writing the dimensions of various quantities in equation 1, we get

 [M¹L¹T⁻²] = 1 [M]ᵃ [LT⁻¹]ᵇ [L]ᶜ

⟹  [M¹L¹T⁻²] = [M]ᵃ [L]ᵇ ⁺ ᶜ [T]⁻ᵇ

Comparing the dimensions of similar quantities on both sides, we get

      a = 1

      b + c = 1 and 

      - 2 = - b ⟹ b = 2

∴ c = 1 - b = 1 - 2 = - 1

∴ a = 1, b = 2 and c = - 1

∴ F = K m v² r⁻¹ = K mv²/r

This is the required expression for the centripetal force.

 

Q10. The velocity  v of water waves depends on the wavelength λ, density of water ρ, and the acceleration due to gravity g. Did use by the method of dimensions the relationship between these quantities. 

 

Soln. Let  v = K λᵃ ρᵇ gᶜ -------(1)

where  K = a dimensionless is constant

Dimensions of the various quantities are

[v] = [LT⁻¹],  [λ] = [L],  [ρ] =  [M¹L⁻³], [g] = [LT⁻²]

Substituting these dimensions in equation (1), we get

[LT⁻¹] = [L]ᵃ  [M¹L⁻³]ᵇ [LT⁻²]ᶜ

∴ [M⁰ L¹T⁻¹] = [Mᵇ Lᵃ⁻³ᵇ⁺ᶜ T⁻²ᶜ]

Equating the powers of M, L and T on both sides, 

b= 0 ; a - 3b + c =1 ; - 2c = - 1

On solving,  a= ½ ; b = 0, c = ½

∴ v = K √(λg)

 

Q11. The frequency "ν" off vibration of a a stretched string depends up on:

a. Its length l

b. Its mass per unit length m and

c. The tension T in the string.

Obtain dimensionally an expression for frequency ν.

 

Soln. Let the frequency of vibration of the string be given by

       ν = K lᵃ Tᵇ mᶜ ----------(1)

where K is a dimensionless constant.

Dimension of the various quantities are

[ν] = [T⁻¹] ; [l] = [L]; [T] =  [M¹L¹T⁻²] ; [m] = [M¹L⁻¹]

Substituting this dimensions in equation 1,  we get

 [T⁻¹] = [L]ᵃ  [M¹L¹T⁻²]ᵇ  [M¹L⁻¹]ᶜ

⟹ M⁰ L⁰ T⁻¹ = Mᵇ ⁺ ᶜ Lᵃ ⁺ ᵇ ⁻ ᶜ T⁻²ᵇ

Equating the dimensions of M, L and T , we get

b + c = 0;  a + b - c = 0; - 2b = - 1

On solving, a = - 1, b = ½, c = - ½

∴ ν = K l⁻¹√(T/m) = (K/l)√(T/m)

 

Q12. The period of vibration of A tuning fork depends on the length l of its prong, density d and Young's modulus Y of its material. Deduce an expression for the period of vibration on the basis of dimensions.